Extensions of Schoen--Simon--Yau and Schoen--Simon theorems via iteration \`{a} la De Giorgi

Abstract

We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of $6$-dimensional (stable minimal) immersions. The key step is an $\epsilon$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension $n\geq 2$, that may have a singular set $\Sigma$ of locally finite $\mathcal{H}^{n-2}$-measure, and that is weakly close to a hyperplane. (In fact, if $\mathcal{H}^{n-2}(\Sigma)=0$, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an $\epsilon$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen--Simon theorem (1981). In both $\epsilon$-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method \`{a} la De Giorgi (from the linear De Giorgi--Nash--Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both $\epsilon$-regularity theorems the method gives explicit constants that quantify the required smallness.)